Arrangements of hyperplanes I: Rational functions and Jeffrey-Kirwan residue

Abstract

Consider the space $R_Δ$ of rational functions of several variables with poles on a fixed arrangement $Δ$ of hyperplanes. We obtain a decomposition of $R_Δ$ as a module over the ring of differential operators with constant coefficients. We generalize to the space $R_Δ$ the notions of principal part and of residue, and we describe its relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of work by L. Jeffreys and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.